Kelly Can't Fail

Original Article ↗ Hacker News Discussion ↗

A mathematical card-betting puzzle based on the Kelly criterion shows that, with ideal continuous bet sizing and perfect knowledge of probabilities, a player can turn $1 into about $9.08 with zero variance, regardless of the order of cards in a fair 52‑card deck. Commenters probe whether this strategy is truly optimal for expected value, explore alternative strategies with the same expectation but higher variance, and connect the result to concepts like universal portfolios, bankroll management, and majority-vote algorithms. Many point out that real-world constraints—discrete bet sizes, rounding, changing probabilities, imperfect information, and limited bankrolls—make pure Kelly-style guarantees impossible outside of toy models.

Optimality and Expected Return

  • Multiple commenters ask whether the Kelly strategy is truly optimal in this specific fixed-deck game.
  • Consensus in the thread: under the specified rules and “sensible” strategies (notably, always betting everything once only one color remains), all such strategies share the same expected return (~9.08× the initial stake).
  • Kelly is highlighted as special because it achieves this expected return with zero variance in the continuous-bet model; alternatives can have the same EV but much higher variance.
  • Some argue that accepting higher variance without higher expected return is just “gambling.”

Alternative Strategies and Intuition

  • Several alternative strategies are discussed, such as:
    • Doing nothing until only one color remains, then going all-in each time.
    • Toy versions with small decks (e.g., 2 red/2 black) to build intuition and inductively generalize.
  • These strategies are shown or simulated to have the same EV as Kelly in this game but different variance profiles.
  • A concise inductive proof is sketched to show the Kelly payoff formula holds regardless of card order.

Dependence, Information, and Assumptions

  • Some question whether card dependence (non-iid draws) breaks Kelly assumptions or allows a better strategy.
  • Others clarify: information gained from flips is independent of the amount bet, and the described Kelly-style rule already updates bets based on the changing composition of the deck.
  • There is debate around independence in real-world analogies (e.g., coin tosses, trading), with some insisting real processes are not iid.

Discrete Stakes, Rounding, and Practical Limits

  • A major subthread notes that the theoretical result assumes infinitely divisible stakes.
  • With cent-level rounding, very long streaks of one color can drive the stake to zero unless the initial bankroll is very large; simulations and APL code explore concrete thresholds.
  • Simple rounding of Kelly bets performs poorly; a dynamic-programming strategy can guarantee about 8.08× with discrete units, less than the continuous 9.08×.
  • Commenters connect this to Martingale-like issues and real-world constraints: finite bet granularity, counterparty solvency, and transaction costs.

Real-World Use and Caveats

  • Kelly’s use in gambling and investing is discussed, with emphasis that:
    • Real bankroll definition and risk tolerance matter.
    • Probabilities are often estimated and non-stationary, so full Kelly can be too aggressive; practitioners tend toward fractional Kelly.
    • Known paradoxes and changing odds (e.g., Proebsting’s paradox) highlight that “Kelly can’t fail” only holds under strict, often unrealistic assumptions.